Asymptotic growth of symbolic multi-Rees algebras

Abstract

We study the question of finite generation of symbolic multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the symbolic multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function is eventually a polynomial. We further prove that the symbolic multi-Rees algebra of monomial ideals is finitely generated with no assumptions on the analytic spread. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour.Proposition 4.1. [13, Lemma 3.4] There exists a vector h ∈ N r such that Ann R M n = Ann R M h for all n ≥ h. In particular, dim M n = dim M h for all n ≥ h.